Stretching Riemannian spherical solar dynamo model from differentia rotation

Abstract

Stretching solar dynamos from differential rotation in a Riemannian manifold setting is presented. The spherical model follows closely a twisted magnetic flux tube Riemannian geometrical model or flux rope in solar physics, presented previously by Ricca [Solar Physics (1997)]. The spherical model presented here present new and interesting feature concerning its connection with spherical steady solar dynamos. One of this new feature is represented by the fact that the by considering poloidal magnetic field component much weaker than its toroidal counterpart, one obtains a stretch dynamo action where the Riemannian solar spherical line element is proportional to differential rotation. This result is obtained also by using the Vainshtein-Zeldovich stretch, twist and fold (STF) method to generate dynamos. One notes that for high magnetic Reynolds of $R{m}=O({10^{7}})$ the dynamo action is present for a corresponding small stretching factor of $K^{2}=1.6$ where $K^{2}=1$ represents the unstretched dynamo. The constant stretched dynamos considered here are shown to be Riemann-flat, where the Riemann curvature tensor vanishes. Solar cycle dynamos are therefore compatible with the Riemann-flat stretching dynamo model from solar differential rotation presented here.